Artificial gravitation effect on spin-polarized exciton-polaritons
AArtificial gravity effect on spin-polarizedexciton-polaritons
E. S. Sedov , , ∗ , A. V. Kavokin , , School of Physics and Astronomy, University of Southampton, SO17 1NJSouthampton, United Kingdom Department of Physics and Applied Mathematics, Vladimir State Universitynamed after A. G. and N. G. Stoletovs, Gorky str. 87, Vladimir 600000, Russia CNR-SPIN, Viale del Politecnico 1, I-00133, Rome, Italy Spin Optics Laboratory, St. Petersburg State University, Ul’anovskaya 1,Peterhof, St. Petersburg 198504, Russia ∗ evgeny [email protected], + [email protected] Abstract
The pseudospin dynamics of long-living exciton-polaritons in a wedged2D cavity has been studied theoretically accounting for the external mag-netic field effect. The cavity width variation plays the role of the arti-ficial gravitational force acting on a massive particle: exciton-polariton.A semi-classical model of the spin-polarization dynamics of ballisticallypropagating exciton-polaritons has been developed. It has been shownthat for the specific choice of the magnetic field magnitude and the ini-tial polariton wave vector the polariton polarization vector tends to anattractor on the Poincar´e sphere. Based on this effect, the switching ofthe polariton polarization in the ballistic regime has been demonstrated.The self-interference of the polariton field emitted by a point-like sourcehas been shown to induce the formation of interference patterns.
Introduction
Exciton-polaritons in 2D semiconductor cavity structures being an equipollentcombination of cavity photons and elementary matter excitations (excitons) andexhibiting properties of both their constituents represent a fascinating objectfor study both from the fundamental point of view and for applications in opto-electronics. Until recently, one of the key constraints for observation of long-range polariton propagation was the extremely short polariton lifetime whichin the best case wouldn’t exceed few tens of picoseconds [1, 2]. Since exciton-polaritons are composite quasiparticles arising due to the strong coupling ofphotons and excitons, they have a finite lifetime that is governed by lifetime ofphotons in the cavity [3]. In recent papers [4, 5] an increase of the polaritonlifetime up to of the order of 100 ps due to the use of specially designed 2Dcavities with ultra-high quality factors has been reported. In these structures,polaritons can move away from the excitation spot by the distances of the orderof a few millimetres in a cavity plane. The possibility of generation of a long-range coherent polariton flow in high-finesse microcavities is very promising forrealization of ultrafast optoelectronic devices.Given the possibility of long-living polaritons excitation, a subject of thecontrol of their spatial dynamics arises. One of the convenient approaches to1 a r X i v : . [ c ond - m a t . o t h e r] F e b anipulation of exciton-polariton dynamical characteristics is through the gen-eration of an external potential of a required configuration. This idea is widelyused in optics, e.g. for observation of optical Bloch oscillations in waveguide ar-rays, where the gradient of optical properties is provided by the linear variationof the effective index of individual guides [6] or by the temperature gradientapplied to the thermo-optic polymer arrays [7]. In this context, the idea ofpolariton acceleration in a cavity with the gradient of the cavity thickness hasbeen discussed in [8] and then developed in [9]. Reduction of the cavity widthleads to a positive shift of the lower dispersion branch (LB) of polaritons. Sincethe latter is concave, to conserve the energy fixed, LB polaritons are forced toreduce their wave vector component in the propagation direction. Their groupvelocity in the same direction decreases as a result. As it has been shown in [9],in samples where the cavity width decreases linearly with one of the in-planecoordinates polaritons propagate on parabolic trajectories. Herewith, contribu-tion of a cavity thickness variation can be described as an effect of an externalconstant force analogous to the gravitational force acting on massive particles.The effect of the gravitational force on quantum systems including atomicsystems and Bose-Einstein condensates has been widely discussed within lastfew decades [10, 11, 12]. The bouncing of quantum wave packets on a hard sur-face under the effect of the gravitational force has been considered theoreticallyin Ref. [11]. The experimental work [12] is devoted to the observation of thebouncing of the Bose-Einstein condensate of Rb atoms off a mirror inducedby a repulsive dipole potential under the gravity. In parallel, the concept ofsimulation of gravitational effects using artificial systems that reproduce par-ticular characteristics of the actual gravitational system in the laboratory hasbeen developed. This concept is known as the analogue gravity [13, 14]. Theanalogue models are mostly based on various condensed-matter systems includ-ing flowing fluids [15, 16] and superfluids [17, 18], uniform [19, 20, 21, 22, 23]and stratified [25, 26] semiconductor and dielectric media. The shaping of lightbelonging to the short-wavelength spectral range of the continuum by the effectof the gravity-like force created by accelerating solitons has been demonstratedin [20]. In Ref. [21] authors have theoretically predicted the optical effect anal-ogous to the mentioned above quantum bouncing known for cold atoms. Thiseffect manifests itself as a multiple reflection of a short optical pulse propagatingin an optical fiber on a refractive index variation created by a co-propagatingcontinuously decelerated soliton. Experimental observation of bouncing of lightin curved optical waveguides has been reported in [22]. In Ref. [23] the effectof the self-induced “artificial gravity” on the interaction of optical wave packetsin the Newton-Schr¨odinger system has been investigated. The nonlinear effectsare responsible for the appearance of the “gravitational” potential in that sys-tem. The effects of gravitational lensing, tidal forces and gravitational redshiftand blueshift have been emulated. The nonlinear optical Newton-Schr¨odingersystem has been investigated as the analogue of a rotating boson star evolutionin Ref. [24].Due to the specificity of the polariton dynamics under the effect of “artificialgravity”, polaritons have been observed to slow down until a complete stop and2hen reverse their propagation direction [9]. The authors of Ref. [9] refer tosuch type of motion as “slow reflection”. This effect has been proposed in [27]as a tool for suppression of the light reflection by the edge of the sample wichis essential for diminishing the parasitic effects caused by interference of polari-tons propagating towards the edge with the reflected ones. In contrast with thenormal reflection, in the case of “slow reflection” the phase of propagating po-laritons does not change abruptly at the turning point but varies continuously.The latter point is essential for understanding of the non-trivial interference ofthe direct and reflected polariton waves, as we show below.Exciton-polaritons are widely discussed as potential candidates for all-opticalinformation transport, storage and processing because of their non-trivial po-larization properties. The cavity polariton polarization is mostly dependenton two factors that are the long-range electron-hole exchange interaction [28]and the polarization splitting of the photonic cavity modes [29]. In particular,the TE-TM splitting of the cavity eigenmodes leads to the splitting of linearlypolarized polariton dispersion branches that induces the precession of the polar-ization vector of propagating polaritons [31, 30]. One can describe the polaritonpolarization using the pseudospin formalism [32, 33, 34]. The polariton TE-TMsplitting effect can be described as the polariton pseudospin precession aroundan effective magnetic field applied in the cavity plane. The non-trivial polaritonpseudospin dynamics has been revealed in a wide range of experimentally ob-served efects including the polarization multistability [35], spin switching [36],optical spin Hall effect [33], etc. In the presence of the external magnetic field B , the polariton pseudospin dynamics becomes even more complicated. For thefield applied in the Faraday geometry ( B is normal to the cavity plane and iscollinear with the incident light wave vector), the exciton-polariton energy bandstructure is enriched with the Zeeman splitting of right- and left-circularly po-larized polariton states [37, 38, 39].In this paper, we consider the pseudospin dynamics of long-living polaritonsin a wedged 2D microcavity under the influence of the “gravitational” force pro-duced by the cavity width gradient in the presence of the external magnetic fieldapplied normally to the cavity plane. The paper is organized as following. Inthe following section we discuss the model describing the polariton polarizationdynamics within the pseudospin formalism. The next section presents the studyof a particular case of polariton propagation in the gradient direction. The lastsection is devoted to a general case of the polariton propagation oblique to theeffective “gravitational” field. In conclusion, we address the importance of po-lariton self-interference and polarization effects in the “slow reflection” regime. Ballistic Motion of Exciton-Polaritons in GradedMicrocavities
The considered geometry of the problem is schematically shown in Fig. 1. Itbasically coincides with that studied experimentally in Refs. [4, 9]. The sample3epresents a semiconductor microcavity with embedded quantum wells that areholders of excitons. The cavity is characterized by a linear variation of the thick-ness in y -direction that causes a polariton energy gradient in the same direction.The exciton-polaritons are supposed to be excited resonantly by the externallaser beam. We shell consider both the regime of excitation by ultrashort laserpulses and the regime of continuous wave pumping. The initial quantum stateof generated polaritons, their polarization, energy and wave vector are supposedto be set by the exciting laser.Propagation of the polaritons in microcavities can be described by the fol-lowing effective Hamiltonian written in the basis of right and left circular po-larizations: (cid:98) H = (cid:98) T + (cid:98) V + (cid:98) H LT + (cid:98) H M , (1)where (cid:98) T = (cid:126) (cid:98) k m ∗ (cid:98) I and (cid:98) V = (cid:126) βy (cid:98) I are the kinetic energy and the potential energyoperators. (cid:98) k = ( (cid:98) k x , (cid:98) k x ) = ( − i∂ x , − i∂ y ) is the polariton momentum operator, m ∗ is the effective mass of polaritons, (cid:98) I is the 2 × Ψ ( t, r ) = (Ψ + ( t, r ) , Ψ − ( t, r )) T , where“+” and “ − ” correspond to the +1 and − z axis. The impact of the “gravitational” force F = − (cid:126) β induced bythe sample thickness gradient is taken into account by introducing the linearvariation of the polariton potential energy in y -direction.The Hamiltonians (cid:98) H LT and (cid:98) H M describe the longitudinal-transverse splittingof linear polarizations and the effect of the normal to the cavity plane exter-nal magnetic field that induces the Zeeman splitting of circular polarizations,respectively. They are given by (cid:98) H LT = (cid:126) (cid:16)(cid:98) Ω x − i (cid:98) Ω y (cid:17)(cid:16)(cid:98) Ω x + i (cid:98) Ω y (cid:17) , (cid:98) H M = (cid:126) (cid:34)(cid:98) Ω z − (cid:98) Ω z (cid:35) . (2)The operators (cid:98) Ω x and (cid:98) Ω y in (2) represent components of the effective mag-netic field (gauge field) operator (cid:98) Ω = (cid:16)(cid:98) Ω x , (cid:98) Ω y , (cid:98) Ω z (cid:17) that affects the polaritonpropagation: (cid:98) Ω x = ∆ LT (cid:16)(cid:98) k x − (cid:98) k y (cid:17) , (cid:98) Ω y = 2∆ LT (cid:98) k x (cid:98) k y , (3)where ∆ LT is the TE-TM splitting constant.In the absence of external magnetic fields, (cid:98) Ω z = 0. The switching of themagnetic field B in normal to the cavity plane direction ( B x,y = 0) removes thedegeneracy in z direction that leads to the appearance of the third componentof the operator (cid:98) Ω : (cid:98) Ω z = µ B gB (cid:126) , (4)where µ B is the Bohr magneton, g is the polariton g-factor and B ≡ | B | .In the simplest model case, the dynamics of a single polariton as well as oneof a polariton wave packet can be considered as a particle-like dynamics of its4center of mass” along the trajectory r c ( t ) = ( x c ( t ) , y c ( t )). The center of massis characterized by the wave vector k c ( t ) = ( k cx ( t ) , k cy ( t )). Let us introduce apolariton wave function in a center-of-mass frame [40, 41, 42]: Ψ ( t, r ) = e iχ ( t ) Φ ( t, r − r c ) , (5)where χ ( t ) is the global phase that arises due to the classical action calculatedalong the center-of-mass trajectory r c . The time-dependent vectors r c and k c obey the classical equations of motion ∂ t r c = (cid:126) m ∗ k c , ∂ t k c = − β e y (6)corresponing to the classical Lagrangian density L = m ( ∂ t r c ) − (cid:126) βy c ; e y isthe unit vector in the direction of y . Here we neglected by the small effectof the LT-splitting on the center-of-mass motion. For the initial conditionsfully determined by the excitation light properties and taken as r = (0 , k = ( k x , k y ) the solutions of (6) are easily found as x c = (cid:126) k x t / m ∗ , y c = (cid:126) k y t / m ∗ − (cid:126) βt (cid:14) m ∗ , k cx = k x and k cy = k y − βt .For a single particle ballistically propagating along the center-of-mass tra-jectory we finally arrive at the set of coupled equations for the wave functionamplitude Φ ( t ) = (Φ − ( t ) , Φ + ( t )): i∂ t Φ ± = ±
12 Ω cz Φ ± + 12 (Ω cx ∓ i Ω cy ) Φ ∓ , (7)where Ω cx,cy,cz characterize the components of the effective magnetic field Ω c affecting the polariton propagation. Ω cx,cy,cz are described by Eqs. (3)–(4)where we substitute (cid:98) k x,y → k cx,cy . The global phase is found as [40, 41] χ ( t ) = (cid:90) t (cid:18) L ( t (cid:48) ) − ∂ t (cid:48) ( k c ( t (cid:48) ) r c ( t (cid:48) )) (cid:19) dt (cid:48) = − (cid:126) βt m ∗ (3 k y − βt ) . (8)Considering the propagation of a single cavity polariton, we can fully charac-terize its polarization (pseudospin) state by the Stokes vector S = ( S x , S y , S z ),where S x,y,z are the intensities of linear (collinear with the original coordinatebasis axes components and diagonal/antidiagonal ones) and circular polariza-tion components, respectively [32]. The vectors Φ and S are linked betweenthemselves by the following set of expressions: S x = 12 (cid:0) Φ + Φ ∗− + Φ ∗ + Φ − (cid:1) , S y = i (cid:0) Φ + Φ ∗− − Φ ∗ + Φ − (cid:1) , S z = 12 (cid:0) | Φ + | − | Φ − | (cid:1) . (9)The squared absolute value of the polariton wave function is dependent on S = (cid:113) S x + S y + S z . For the fully polarized wave S remains unchanged duringthe polariton propagation. Using this fact, in further discussions we express S x,y,z in the units of S ; | S x,y,z | ≤
1. Finally we obtain the following equation5haracterizing the dynamics of the Stokes vector of the particle propagatingalong the center-of-mass trajectory [33]: d S dt = Ω c × S , (10)It is important to underline that in contrast with the problem consideredin Ref. [33], the absolute value and the orientation of the vector Ω c actingon a specific ballistically propagating polariton becomes time- and coordinate-dependent due to the “effective gravity”. The polarization components oscillatewith a frequency Ω c = | Ω c | that is changing with the particle propagation.Below we list the parameters used for the numerical modelling. Following [9],we consider polaritons possessing the effective mass of m ∗ = 5 × − m e with m e being the vacuum electron mass. The LT-splitting constant is taken ∆ LT =11 . µ eV × µ m . We consider a sample which provides the “gravitational” force F = − . µ eV /µ m. Polariton g-factor is g = 0 .
1. The initial wave number k can be tuned by changing the angle of incidence of the pump. Polariton Propagation in the Direction of the Thick-ness Gradient
Single Polariton Propagation
Let us first consider polaritons propagating in the “gravitational” force vectordirection ( y -direction). To control the polarization state of polaritons, we havethree external parameters. The first one is the “gravitational” force strengthgoverned by the parameter β . In the absence of external magnetic fields ( B = 0)the “gravitational” force doesn’t affect k x component of the wave vector thusit remains unchanged during the polariton propagation. In the considered case k x = 0, according to Eq. (3) we have Ω cy = Ω cz = 0, so that, Eq. (10) yields ∂ t S x = 0. Thus for the initial conditions S y = S z = 0 and S x = ± ± , ,
0) on the Poincar´e sphere.For the initial conditions S y (cid:54) = 0 or/and S z (cid:54) = 0 and | S x | < S x remains constant while S y,z components oscillate in the range( − (cid:112) − S x , (cid:112) − S x ). The solution of Eq. (10) can be easily found analyti-cally as S x = S x , S y = S y cos [ κ ( t )] + S z sin [ κ ( t )] , S z = S z cos [ κ ( t )] − S y sin [ κ ( t )] , (11)where the propagation factor is κ ( t ) = ∆ LT t (cid:2) k y − k y βt + β t (cid:14) (cid:3) . The en-ergy transfer between the circular and the diagonal linear polarization modes isclearly seen. The polariton Stokes vector describes a circular trajectory on thePoincar´e sphere at S x = S x in this case.Another polarization dynamics control parameter is the initial wave vector k in the xy plane. Slightly changing the value of k one can enhance or diminish6he impact of the “gravitational” force on the dynamics of the polarizationcomponents.The third control parameter is an external magnetic field perpendicular tothe cavity plane. It makes S x time-dependent and allows for manipulating theoutput polarization when keeping k unchanged.The color maps in Fig. 2 illustrate the dynamics of the polariton polarizationcomponents S x,y,z as functions of the initial wave number k for a number ofvalues of the external magnetic field magnitude B (Fig. 2(a)) and on the valueof B for different k (Fig. 2(b)). Similar color maps are presented in Fig. 3 forthe initial circular polarization with S z = 1, S x,y = 0. The inclined broad-dashed (left) lines in Figs. 2(a) and 3(a) indicate the turning times, t = k y /β ,when polaritons reach the furthest point in y direction for the given k . Atthis moment k y ( t ) = 0. The inclined narrow-dashed (right) lines indicate thetime t = 2 t when polaritons return to the starting point r c ( t ) = (0 , k =3 . µ m − . The vertical dashed lines in Figs. 2(b) and 3(b) correspond to t (left) and t (right).Let us now consider in detail the main peculiarities of the polariton polar-ization dynamics illustrated by Figs. 2–3. First, the central colorless horizontalstripes in the left panels in Figs. 2(b) and 3(b) corresponding to B ≈ S x remains unchanged during the propagation in the absence of an externalapplied magnetic field.Second, for the strong enough magnetic field, B (cid:29) cz sufficientlyexceeds Ω cx,cy , the oscillation frequency Ω c tends to a constant value, Ω c → Ω cz = const . For the cases of both linear and circular initial polarizations anda large magnetic field, S x polarization component decreases at long times, seethe left lower panels in Figs. 2(b) and 3(b). This can be understood in termsof the interplay between the dynamics of the effective magnetic field and thedynamics of of the polariton Stokes vector that precesses around this field.Even stronger the interplay dynamical effects manifest themselves in the in-termediate regime of Ω cz ∼ Ω cx,cy . This regime is characterized by the enhance-ment of the S x component of the Stokes vector accompanied by suppression ofthe other components, S y and S z . The latter is visualised as pale stripes inFigs. 2 and 3. Herewith, for the given B , there is a discrete spectrum of k thatcorresponds to the appearance of such stripes. The value of k increases withthe increase of the index of the stripe n approximately quadratically k n ∝ n .Having in mind that Ω cx ∝ k , this is a signature of a the resonant parametriccharacter of the formation of closed trajectories at the surface of the Poincar´esphere as will be discussed below.To analyse principal peculiarities of the discovered non-trivial polarizationdynamics, let us track the corresponding phase trajectories on the Poincar´esphere. Examples of the resulting trajectories are shown in Fig. 4(a) for thelinear ( S y = 1, upper panels) and circular ( S z = 1, lower panels) initial polar-izations. The considered states are characterized by low-amplitude oscillations s x,y,z ( t ) of the polarization parameters at t → ∞ around some stationary point(attractor) with S x ( ∞ ) different from zero. We represent the Stokes parameters7s S x,y,z ( t ) = (cid:104) S x,y,z ( t ) (cid:105) + s x,y,z ( t ), where (cid:104) S x,y,z ( t ) (cid:105) characterize slowly chang-ing mean values, |(cid:104) S x,y,z (cid:105)| (cid:29) | s x,y,z | and ∂ t (cid:104) S x,y,z (cid:105) (cid:28) ∂ t s x,y,z . Substitutingthese expansions in (10), we finally obtain that (cid:104) S y (cid:105) = 0 while the rest param-eters are related between themselves as (cid:104) S z (cid:105) = Ω cz Ω cx (cid:104) S x (cid:105) . Since Ω cx grows withtime as t while Ω cz remains constant, for large enough time (cid:104) S z (cid:105) tends to zero.Whereas the parameters S x,y,z are linked with each other trough S , the (cid:104) S x (cid:105) polarization component tends to 1 for t → ∞ . Consequently, the discoveredclosed trajectories on the Poincar´e sphere correspond to the oscillations of theStokes vector in the S y S z plane around the point ( (cid:104) S x (cid:105) , (cid:104) S y (cid:105) , (cid:104) S z (cid:105) ) =(0 , , S y = ±
1) or the circular polarization state ( S z = ±
1) to thelinear polarization state ( S x (cid:39) ± S given byEq. (10). This vector characterizes precession of the polariton Stokes vectoraround the effective magnetic field Ω = (Ω cx , Ω cy , Ω cz ). For the considered casewhere Ω cy = 0, Eq. (10) yields | ˙ S | = Ω cx (cid:0) S y + S z (cid:1) + Ω cz (cid:0) S x + S y (cid:1) − cx Ω cz S x S z . (12)The temporal behavior of | ˙ S | is illustrated in Fig. 4(b) for a number ofparticular cases. It is clearly seen that for the parameters corresponding to thepronounced polarization switch trajectories in Fig. 4(a) there is a significantreduction of | ˙ S | (solid lines in Fig. 4(b)) at the times around t . In the limitingcase when | ˙ S | tends to zero at large time, the trajectories degenerate to the point.The closest to this regime trajectories are shown in the upper panels of Fig. 4(a)while the corresponding dynamics of | ˙ S | is shown in Fig. 4(b) (bold black solidcurve). The exact solutions of the equation | ˙ S | = 0 are inachievable as S x,y change with time. Nevertheless, if this equation is satisfied even approximatelyfor t ∼ t , the closed trajectories (attractors) may be seen on the surface of thePoincar´e sphere. | ˙ S | slowly increases with time at t → t in this regime. Self-Interference of Polarized Polariton Waves
Let us now consider the different excitation regime, namely the resonant cwexcitation of polaritons in the spot characterized by r c (0) = (0 ,
0) and k x = 0, k y = k . Obviously, in this case the polariton polarization dynamics is enrichedby the effects coming from the self-interference of the polariton wave. Since thepseudospin formalism is not applicable for the description of the interferenceeffects, here we deal directly with the polariton field amplitudes Ψ ± governedby Eqs. (5)–(8).Figure 5 shows the interference intensity patterns as functions of B for thelinear, S y = 1 and S x,z = 0, (a) and the circular, S z = 1 and S x,y = 0, (b)initial polarizations. The pictures are rather different from the interferencepatterns of an incident and a reflected waves in the case of a conventionalcoherent reflection of light by a mirror. They mostly demonstrate the Airy-like8rofile in y direction. In addition, in contrast with the conventional reflectionby the mirror, the phase of the backward wave changes continuously and doesnot show a discontinuity with the incident wave phase at the mirror surface.This is a signature of the “slow mirror” effect [9].In the particular case corresponding to the purely polarized initial wave,e.g. the circularly polarized wave considered in Fig. 5(b), and in the absenceof the magnetic field, B = 0, the interference picture can be described an-alytically. To do so, we formally split the ballistically propagating polaritonwave into two waves, namely the forward wave propagating “uphill” (towardhigher potential energies and narrower cavity widths) Ψ (up) and the back-ward wave propagating “downhill” (in the opposite direction) Ψ (dn) denoting Ψ (up,dn) = (Ψ (up,dn)+ , Ψ (up,dn) − ) T . It is convenient to introduce the “time offlight” parameter τ ≡ τ ( y ) being a time of propagation of light emitted by asource at y = 0 until the point having a coordinate y . Note that light passestwice through each point except those with y > y max , where y max is given by y max = (cid:126) k y (cid:14) βm ∗ . But we mean the shortest time of flight from the sourceto the point y .To consider Ψ (up) and Ψ (dn) as co-propagating waves, we shift the originof the t axis to t and make the time reversion, t → − t , i.e. we re-introducethe backward wave as Ψ (dn) ( t ) → Ψ (dn) ( t − t ). Starting with the solutionsΨ + = cos [ κ ( t ) /
2] exp[ iχ ( t )] and Ψ − = i sin [ κ ( t ) /
2] exp[ iχ ( t )] with κ ( t ) givenabove we finally obtain the resulting interference pattern as I y> = 2 + cos [ k cy ( τ ) d p ( τ )] + cos [ k cy ( τ ) d m ( τ )] , (13)where the effective coordinates d p,m ( τ ) are given by d p,m ( τ ) = − (cid:104) ( k cy ( τ )) (2 m ∗ ∆ LT ∓ (cid:126) ) ± (cid:126) k y (cid:105) / m ∗ β, and the time of flight parameter changes from τ = 0 to t to cover the distancefrom y = 0 to y max . The expression (13) describes the trajectory in Fig. 5(b)at B = 0.In the presence of the external magnetic field the interference patterns pro-duced by self-intefering polaritons can not be described by a simple analyticalexpression. The magnetic field effect on the interference patterns is presented inFig. 5. It is obvious from the figures that the magnetic field not only introducesa phase shift to the interfering waves but also affects the spreading of the wavesin space. One should also mention that the patterns in Fig. 5(b) are asymmetricwith respect to B = 0, and the interference picture is reversed in the case of theopposite initial polarization, S z = − ropagation of Polaritons in the Oblique to theGradient Direction Single Polariton Propagation
In practice, we have one more external control parameter in our problem that isthe shooting angle θ , the angle between the initial polariton wave vector and x -axis. In this case, k x and the related parameter Ω cy differ from zero. The maindifference from the case considered above is that now the particle propagatesalong a parabolic trajectory, hence the forward and the backward branches areseparated by a distance (cid:126) k m ∗ β | tan[ θ ] | [ θ ] in the x direction at the latitude x = 0.In [9] the spatial separation between forward and backward going polaritonshelped measuring the long-living polariton lifetime since it allowed eliminatingthe spurious radiation resulting from the reflection of the excitation laser.Figure 6(a) demonstrates the spatial distribution of the polariton polar-ization components S x,y,z in the cavity plane for different initial conditionsin the absence of the external magnetic field, B = 0. The initial polariza-tions are taken S y = 1 for the upper panels, S x = 1 for the middle panelsand S z = 1 for the lower panels. The other initial polarization componentsare taken equal to zero in all cases. The polarization components S x,y,z aremultiplied by the spatially dependent wave packet intensity of the polaritonfield. The incident wave packet is governed by the shape of the Gaussian form E ∼ exp[ − ( y − y c ) (cid:14) y w ] with y w = 0 .
04 mm and the time-dependent “centerof mass” coordinates ( x c ( t ) , y c ( t )). The upper panels in Fig. 6(a) demonstratethe transformation of the diagonal linear polarization S y = 1 to the antidiago-nal one, S y = −
1. The Stokes vector evolves from S y = 1 to S y = − S x = 1 state. The component S z is close to zero in this case. Themiddle and the lower panels demonstrate the inversion of the linear (middle)and circular (lower) polarizations, as one can see comparering the polarizationpatterns before and after the turning point. The Pulsed Excitation of Exciton-Polaritons by a Point-likeSource
We now consider the polarization dynamics of the cylindrical polariton wavepacket emitted by a point-like source placed in the point (0 , S x = S y = 0 and S z = 1.Figures 6(b) represent the parametric dependencies of the polariton polarizationcomponents on the spatial coordinates: x and the shifted coordinate ¯ y = y + (cid:126) βt m ∗ . The black dashed circles in Fig. 6(b) correspond to t = 40 ps, 80 ps and120 ps (counting from the inner to the outer circle).The upper row panels illustrate the trivial case without the “gravitational”force ( β = 0) and the magnetic field ( B = 0). It is the case discussed in Ref. [33]and showing the signature of the optical spin Hall effect. In the middle rowpanels, the “gravitation” is switched on, (cid:126) β = 10 . µ eV /µ m. In this case, the10olarization patterns “fall down” in time and the symmetry (or anti-symmetry)of the patterns with respect to the point moving with the velocity − (cid:126) βt / m ∗ down the y direction is conserved. In the lower row panels, the magnetic field B = 3 T has been added as well. The magnetic field B breaks the patternsymmetry and induces the rotation of the patterns clockwise for B <
B >
0. Since we do not consider a continuous pump here, theinterference effects are absent.
Self-Interference of a Polariton Flow Excited by a Point-likeSource in the CW Regime
In contrast with the previous subsection, here we consider the case where thepoint (0 ,
0) is a source of continuous polariton waves. Figure 7 demonstratesthe real-space intensity patterns appearing as a result of the self-interference ofthe polariton wave initially emitted as a cylindrical wave. The main peculiarityof the patterns is that they are limited in + y direction. The “slow mirror”boundary corresponds to the envelope of the classical center-of-mass trajectoriesof the polaritons emitted by the point-like source. The boundary possesses aparabolic shape obeying the equation y SM = (cid:126) k βm − βm (cid:126) k x . (14)Another peculiarity is that the interference fringes are concentrated in theupper part of the interference picture, while the lower part is characterizedby the uniform intensity distribution. Above we obtained the analytical ex-pression (13) for the interference fringes at y > y < k dn0 = (0 , − k ). Another wave that we refer toas the reflected wave is one emitted uphill with the wave vector k up0 = (0 , k ),then reflected by the “slow mirror” and returned to the starting point after t = t with the geometric phase shift. Based on the known solutions we findthat at y < y : I y< =2 + cos (cid:2) k (∆ LT + (cid:126) /m ∗ ) (cid:14) β (cid:3) + cos (cid:2) k (∆ LT − (cid:126) /m ∗ ) (cid:14) β (cid:3) .The effect of the external magnetic field on the interference patterns isdemonstrated in Fig. 7(b). It introduces a phase shift in the interfering wavesthat results in the shift of the intensity fringes in the upper part of the interfer-ence picture and in the change of the total intensity in the lower part. Conclusions
We have considered the dynamics of exciton-polaritons propagating over largedistances in wedged microcavities in the presence of the TE-TM splitting and11xternal magnetic fields. The cavity thickness gradient playing role of the arti-ficial gravity strongly affects the TE-TM splitting and the speed of propagatingpolaritons. We demonstrated theoretically the formation of polarization pat-terns in the real space due to the optical spin Hall effect and “slow reflection”.We have demonstrated the possibility of manipulating the polariton polar-ization state by choosing the excitation laser incidence angle and the magnitudeof the magnetic field. Interestingly at the specific combinations of the magneticfield and the initial polariton wave vector the polariton Stokes vector tends toan attractor on the surface of the Poincar´e sphere.The “gravitational” force compelling polaritons to change both magnitudeand direction of their wave vector is the cause of another fascinating effect thatis the self-interference of a spin-polarized polariton field. We show that for apoint-like polariton source in the wedged microcavity the “slow mirror” has aparabolic shape determined by the “gravitational” force and the initial polaritonwave number. The effect of the external magnetic field on the interferencepatterns can be considered as well. The effects discussed here may be used infuture polariton spin transistors and logical gates.
Acknowledgments
This work was supported by the EPSRC Programme grant on Hybrid Polari-tonics No. EP/M025330/1. The work of E.S.S. was supported by the RussianFoundation for Basic Research Grants No. 16-32-60104, No. 15-59-30406, andby grant of President of Russian Federation for state support of young Rus-sian scientists No. MK-8031.2016.2. A.V.K. acknowledges the partial supportfrom the HORIZON 2020 RISE project CoExAn (Grant No. 644076). E.S.S.acknowledges Dr. I. Yu. Chestnov for valuable advices and fruitful discussions.
Contributions
A.V.K. initiated the study and proposed the theoretical model. E.S.S. con-ducted the theoretical simulations and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
References [1] Kasprzak, J. et al.
Bose-Einstein condensation of exciton polaritons.
Nature , 409–414 (2006).[2] Amo, A., et al.
Polariton superfluids reveal quantum hydrodynamic soli-tons.
Science , 1167–1170 (2011).123] Keeling, J. & Berloff, N. G. Exciton-polariton condensation.
ContemporaryPhysics , 131–151 (2011).[4] Nelsen, B. et al. Dissipationless Flow and Sharp Threshold of a PolaritonCondensate with Long Lifetime.
Phys. Rev. X , 041015 (2013).[5] Steger, M. et al. Long-range ballistic motion and coherent flow of long-lifetime polaritons.
Phys. Rev. B , 235314 (2013).[6] Morandotti, R., Peschel, U. & Aitchison, J. S. Experimental Observationof Linear and Nonlinear Optical Bloch Oscillations. Phys. Rev. Lett. ,4756–4759 (1999).[7] Pertsch, T., Dannberg, P., Elflein, W., and Br¨auer, A. Optical Bloch Os-cillations in Temperature Tuned Waveguide Arrays. Phys. Rev. Lett. ,4752–4755 (1999).[8] Sermage, B., Malpuech, G., Kavokin, A. V. & Thierry-Mieg, V. Polaritonacceleration in a microcavity wedge. Phys. Rev. B , 081303(R) (2001).[9] Steger, M., Gautham, C., Snoke, D. W., Pfeiffer L. & West, K. Slow reflec-tion and two-photon generation of microcavity exciton-polaritons. Optica , 1–5 (2015).[10] Saba, C. V. et al. Reconstruction of a Cold Atom Cloud by MagneticFocusing.
Phys. Rev. Lett. , 468–471 (1999).[11] Gea-Banacloche, J. A quantum bouncing ball. Am. J. Phys. , 776–782(1999).[12] Bongs, K. et al. Coherent Evolution of Bouncing Bose-Einstein Conden-sates.
Phys. Rev. Lett. , 3577–3580 (1999).[13] Visser, M., Barcel´o, C. & Liberati, S. Analogue Models of and for Gravity. General Relativity and Gravitation , 1719–1734 (2002).[14] Barcel´o, C., Liberati, S., Visser, M. Analogue Gravity. Living Rev. Relativ. , 3–159; 10.12942/lrr-2011-3 (2011).[15] Unruh, W. G. Experimental black-hole evaporation?, Phys. Rev. Lett. ,1351–1353 (1981).[16] Weinfurtner, S., Tedford, E. W., Penrice, M. C. J., Unruh, W. G. &Lawrence, G. A. Measurement of stimulated Hawking emission in an ana-logue system, Phys. Rev. Lett. , 021302 (2011).[17] Jacobson, T. A. & Volovik, G. E. Effective spacetime and Hawking radi-ation from moving domain wall in thin film of He-3-A.
Pis’ma Zh. Eksp.Teor. Fiz. , 833–838 (1988). 1318] Jacobson, T. A. & Volovik, G. E. Event horizons and ergoregions in He-3, Phys. Rev. D , 064021 (1998).[19] Philbin, T. G. et al. Fiber-optical analogue of the event horizon,
Science , 1367–1370 (2008).[20] Gorbach, A. V. & Skryabin, D. V. Light trapping in gravity-like potentialsand expansion of supercontinuum spectra in photonic-crystal fibres.
NaturePhotonics , 653-657 (2007).[21] Gorbach, A.V. & Skryabin, D.V. Bouncing of a dispersive pulse on an accel-erating soliton and stepwise frequency conversion in optical fibers. OpticsExpress , 14560–14565 (2007).[22] Della Valle, G. et al. Experimental Observation of a Photon Bouncing Ball.
Phys. Rev. Lett. , 180402 (2009).[23] Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C. & Segev, M. Opti-cal simulations of gravitational effects in the Newton-Schro¨odinger system.
Nature Physics , 872–878 (2015).[24] Roger, T. et al. Optical analogues of the Newton-Schr¨odinger equation andboson star evolution.
Nature Commun. , 13492 (2016).[25] Sedov, E. S., Iorsh, I. V., Arakelian, S. M., Alodjants, A. P. & Kavokin,A. Hyperbolic Metamaterials with Bragg Polaritons. Phys. Rev. Lett. ,237402 (2015).[26] Sedov, E. S. et al.
Hyperbolic Metamaterials Based on Bragg PolaritonStructures.
JETP Letters, , 62–67 (2016).[27] Gao, T. et al.
Polariton condensate transistor switch.
Phys. Rev. B ,235102 (2012).[28] Maialle, M. Z., de Andrada e Silva, E. A. & Sham, L. J. Exciton spindynamics in quantum wells. Phys. Rev. B , 15776–15788 (1993)[29] Kavokin, A. Spin Effects in Exciton-Polariton Condensates in “ ExcitonPolaritons in Microcavities. New Frontiers ” (ed. Timofeev, V., Sanvitto,D.) 233–244 (Springer-Verlag, Berlin Heidelberg, 2012).[30] Panzarini, G. et al.
Exciton-light coupling in single and coupled semicon-ductor microcavities: Polariton dispersion and polarization splitting.
Phys.Rev. B , 5082–5089 (1999).[31] Shelykh, I. A., Rubo Yu. G. & Kavokin, A. V. Renormalized dispersion ofelementary excitations in spinor polariton condensates. Superlattices andMicrostructures , 313–320 (2007).[32] Kavokin, A., Baumberg, J. J., Malpuech, G. & Laussy, F. P. Microcavities (Oxford University Press, 2007). 1433] Kammann, E. et al.
Nonlinear Optical Spin Hall Effect and Long-RangeSpin Transport in Polariton Lasers.
Phys. Rev. Lett. , 036404 (2012).[34] Kavokin, K. V., Shelykh, I. A., Kavokin, A. V., Malpuech, G. & Bigenwald,P. Quantum Theory of Spin Dynamics of Exciton-Polaritons in Microcavi-ties.
Phys. Rev. Lett . , 017401 (2004).[35] Gippius, N. A. et al. Polarization Multistability of Cavity Polaritons.
Phys.Rev. Lett. , 236401 (2007).[36] Amo, A. et al. Exciton-polariton spin switches.
Nature Photonics , 361–366 (2010).[37] Kochereshko, V.P. et al. Lasing in Bose-Fermi mixtures.
Sci. Rep. , 20091(2016)[38] Solnyshkov, D. D. et al. Magnetic field effect on polarization and disper-sion of exciton-polaritons in planar microcavities.
Phys. Rev. B , 165323(2008)[39] Morina, S., Liew, T. C. H. & Shelykh, I. A. Magnetic field control of theoptical spin Hall effect. Phys. Rev. B , 035311 (2013).[40] Takagi, S. Quantum Dynamics and Non-Inertial Frames of Reference. III. Progr. Theor. Phys. , 783–798 (1991).[41] Meister, M. et al. Efficient description of Bose-Einstein condensates intime-dependent rotating traps.
Advances In Atomic, Molecular, and Opti-cal Physics , 375–438 (2017).[42] Garc´ıa-Ripoll, J. J., P´erez-Garc´ıa, V. M. & Vekslerchik, V. Construc-tion of exact solutions by spatial translations in inhomogeneous nonlinearSchr¨odinger equations. Phys. Rev. E , 056602 (2001).15igure 1: (Color online) The schematic of the considered problem. The sam-ple represents a 2D electrodynamical microcavity with embedded ensemble ofsemiconductor quantum wells. The width of the cavity in z direction gradu-ally decreases along y axis. Polaritons are injected in the sample by a laserbeam inclined to the cavity plane. By manipulation the angle of inclination,one possible to control the polariton kinetic energy in the sample in xy plane.An external magnetic field B is applied to the system normally to the cavityplane.Figure 2: (Color online) The dynamics of the polariton polarization com-ponents S x,y,z in the case of polariton propagation along the gradient of thethickness as a function of the initial wave number k (a) for B = 0 T (the upperpanels) and 1 . B (b) for k = 1 . µ m − (the upper panels) and 3 . µ m − (the lower panels)The left upper panel in (a) shows schematically the polariton trajectory. Theinitial polarization is taken diagonal linear with the initial conditions S y = 1, S x,z = 0. The parameters used for this calculation are given in the text of thepaper. 16igure 3: (Color online) The dynamics of the polariton polarization com-ponents S x,y,z in the case of polariton propagation along the gradient of thethickness as a function of the initial wave number k (a) for B = 0 T (the upperpanels) and 2 . B (b) for k = 1 . µ m − (the upper panels) and 3 . µ m − (the lower panels). Theleft upper panel in (a) shows schematically the polariton trajectory. The initialpolarization is taken circular with the initial conditions S z = 1, S x,y = 0. Theparameters used for this calculation are the same as in Fig. 2.17igure 4: (Color online) (a) Projections of the Poincar´e sphere and the trajec-tories of the polariton Stokes vector. In the upper panels B = 1 . S y = 1, S x,z = 0. In the lower panels B = 2 . S z = 1, S x,y = 0. The blue-colored partsof the curves show the Stokes vector trajectories from t = 0 to t . The red-colored curves show the Stokes vector evolution after t . (b) The absolute valueof the time derivative of the polariton Stokes vector calculated as a functionof time. The dependence is shown in the interval 0 ≤ t ≤ t . The value of˙ S ( t ) is normalized to ˙ S (0). The bold black curves correspond to the diagonallinear initial polarization with S y = 1 while the thin blue curves correspond tothe circular initial polarization with S z = 1. The initial wave number is taken k = 3 . µ m − . The values of B are taken 0 . . . . y axis as functions of the magnitude of theexternal magnetic field B . (a) corresponds to the diagonal linear initial polar-ization, S y = 1, S x,z = 0, (b) corresponds to the circullar initial polarization, S z = 1, S x,y = 0. The initial wave number is taken k = 1 µ m − .18igure 6: (Color online) (a) Polariton polarization components S x,y,z of bal-listically propagating exciton-polaritons at an oblique angle. The initial polar-ization is taken diagonal linear with S y = 1 for the upper panels, linear with S x = 1 for the middle panels and circular with S z = 1 for the lower panels.The magnetic field B = 0, the initial wave number is taken k = 3 . µ m − , theshooting angle is taken 45 ◦ ( k x = k y = k / √
2) for all the panels. (b) Polariza-tion patterns resulting from the pulsed excitation of polaritons by a point-likesource calculated at different values of β and B . For the upper panels, β = 0, B = 0; for the middle panels, (cid:126) β = 10 . µ eV /µ m, B = 0; for the lower panels, (cid:126) β = 10 . µ eV /µ m, B = 3 T. The initial polarization is taken circular with theinitial conditions S x,y = 0, S z = 1. For the clarity of representation, in themiddle and lower panels we introduces a new dynamically shifted coordinate¯ y = y + (cid:126) βt (cid:14) m ∗ . Dashed circles (from the inner to the outer) correspond to t = 40 ps, t = 80 ps and t = 120 ps.Figure 7: (Color online) Patterns resulting from the self-interference of thecircularly polarized polariton field emitted by a point source placed in (0 , B = 0 (a) and B = 3 T (b). The initial wave number is taken k =1 µ m −1